模擬輸出分析--重疊分批變異數法

Methods of Overlapping Batch Variances for Simulation Output Analysis

葉嘉舜(Chia-Shuen Yeh)

指導教授 ： 楊維寧 ； 陳慧芬

中原大學/工學院/工業工程研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200392

摘要

摘 要
模擬輸出分析—重疊分批變異數法
葉嘉舜

本論文是在探討樣本變異數之變異數的估計問題。當系統分析者利用模擬實驗得到一組輸出資料來估計母體變異數時，通常將點估計設為樣本變異數。更進一步地，我們還會想要估計樣本變異數的變異數，以用來計算樣本變異數的信賴區間、有效位數、準確度等。因此如何估計樣本變異數之變異數為一重要的課題。如果輸出資料互相獨立，估計變異數的方法很簡單，但是在隨機系統的模擬實驗中，經常輸出資料是有相關性的。例如在等候線中，一工件的待加工時間會影響排在其後面之工件的待加工時間。給定一串資料為工件的加工時間，若欲估計的系統績效指標為工件的待加工時間的變異數，則系統績效指標的點估計為這些加工時間的樣本變異數。因為這些加工時間是互相關的，因此如何估計樣本變異數之變異數就變的複雜多了。

我們使用分批變異數法來解決此估計問題，依照分批的方式，分批法可分成不重疊和重疊分批兩種。在分批法的文獻中有很多是探討分批平均數法，但卻少有文獻探討分批變異數法。本論文針對不重疊(NBV)與重疊分批變異數(OBV)法的統計特性加以討論。在線性過程(linear process)裡，我們推導NBV估計量的偏誤及變異數，並加以推廣計算OBV估計量的偏誤及變異數，更進一步地，我們討論OBV估計量之最佳批量的大小。

此外，假若資料量大小n是事先未知的（如線上作業方式）或記憶體不足而無法將所有的資料存入時，我們提出動態不重疊分批變異數(DNBV)法並加以討論DNBV估計量裡的TNBV估計量與PNBV估計量的統計特性。

關鍵詞：動態分批平均數法，不重疊分批平均數法，重疊分批平均數法，最佳批量，樣本變異數，模擬輸出分析

關鍵字

重疊分批平均數法 ； 樣本變異數 ； 不重疊分批平均數法 ； 模擬輸出分析 ； 動態分批平均數法 ； 最佳批量

並列摘要

ABSTRACT

Methods of Overlapping Variances for Simulation Output Analysis

Chia-Shuen Yeh

We consider the case that the performance measure of interest is a population variance, which can be estimated by the sample variance (with denominator being the number of data) based on a set of identically distributed but correlated data. The research problem is to estimate the variance of the sample variance, for purposes such as constructing confidence intervals.

The batching method for output analysis with correlated data is easy to implement and requires only a single long run in the simulation experiment. Intensive research has been devoted to the problem of estimating the variance of the sample mean, but little to the sample variance, for the case that the interested performance measure is a mean.

The batching method estimates the variance of the sample variance by dividing the observations into several batches. Sample variances, called
batch variances, of batched data are computed for each batch. The variance estimator for the sample variance is therefore a function of the batch variances. Depending on whether the batches overlap or not, the estimator is defined ferently as nonoverlapping batch variance (NBV) and overlapping batch variance (OBV) estimators, respectively.

By viewing the sample variance as a sample mean of the quadratic terms, we show that the asymptotic results for the method of batch variances work in the same way as those for the method of batch means with the output data defined as the quadratic terms. The asymptotic results consist of three parts: convergence rate,
constant multiplier, and data properties. We show that both methods have the same convergence rate and constant multiplier and that the data properties for the method of batch variances
are the analogous properties for batch means whose data are quadratic terms. The proofs are provided for NBV estiamtors of linear processes. For OBV estimators, we assume that the asymptotic ratios between OBV and NBV are the same as those for batch means; hence, the OBV estimator has the proposed asymptotic results. The asymptotic bias for the first-order-moving-average process is computed as an evidence.

In addition, we consider the method of dynamic NBVs (DNBVs) for the cases that the computer storage is small compared to the number of observations and that the number of observations is unknown in advance. We propose an approach of storing the batch variances dynamically and show that the asymptotic results for batch-mean estimators remain applicable to DNBV estimators.

Keywords: Dynamic batch means; nonoverlapping
batch means; optimal batch size; overlapping batch means; samplevariance; simulation output analysis

並列關鍵字

sample variance ； optimal batch size ； overlapping batch means ； dynamic batch means ； simulation output analysis ； nonoverlapping batch means

參考文獻

[2] Arnold, S. F. "Mathematical Statistics". Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1990.

[3] Bickel, P. J. and Doksum, K. A. "Mathematical Statistics" (Vol. 1, 2nd ed.). Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 2001.

[5] Brillinger, D. R. "Time Series: Data Analysis and Theory". New York: Holt, Renihart and Winston, Inc., 1975.

[8] Conway, R. W. "Some tactical Problems in digital simulation". Management Science, 24, 47--61, 1963.

[9] David, H. A. "Bias of S^2 under dependency". The American Statistician, 39(3), 201, 1985.